Laplacian of Gaussian (LoG)

The Laplacian of Gaussian, or LoG, combines Gaussian smoothing with a second-order derivative. It locates an edge at a zero crossing of the filtered response.

Main idea

The two LoG response lobes show the intensity transition. The zero crossing between them marks the edge position.

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Processing Sequence

flowchart LR

I["Input edge"] --> G["Gaussian smoothing"]
G --> L["Laplacian response"]
L --> Z["Zero crossing<br/>edge position"]

Smoothing reduces rapid noise changes before the second derivative is measured.

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Inverted Gaussian Convention

Using an inverted Gaussian sign convention:

$$h(r)=-\exp \left(-\frac{r^2}{2{\sigma }^2}\right),r^2=x^2+y^2$$

At the centre, $h(0)=-1$. At radial distance $r=\sigma$:

$$h(\sigma )=-e^{-1/2}\approx -0.607$$

Symbol	Meaning
$r$	Radial distance from the filter centre
$\sigma$	Gaussian scale
$h(r)$	Inverted Gaussian response

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Laplacian of the Gaussian

The two-dimensional Laplacian is:

$${\nabla }^{2}h(r)=\frac{{\partial }^{2}h}{\partial x^2}+\frac{{\partial }^{2}h}{\partial y^2}=-\left(\frac{r^2-2{\sigma }^2}{{\sigma }^4}\right)e^{-r^2/(2{\sigma }^2)}$$

This response has:

• A positive central lobe
• Negative side lobes
• A zero-crossing ring at radius $r=\sqrt{2}\sigma$
• A negative ring near radius $r=2\sigma$

The filter can be applied in either equivalent order:

$$f\ast {\nabla }^{2}h={\nabla }^2(f\ast h)$$

This means that convolving with the LoG kernel is equivalent to smoothing with $h$ and then applying the Laplacian.

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How an Edge Produces a Zero Crossing

A step edge gives a double-lobed LoG response:

1. One lobe appears on one side of the intensity transition
2. The response changes sign at the transition
3. The second lobe appears on the other side
4. The sign change gives the edge position

Do not select the LoG peaks as the edge position

The peaks show the two sides of the response. The zero crossing between them locates the edge.

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Effect of $\sigma$

Choice of $\sigma$	Main effect
Smaller $\sigma$	Detects finer changes and retains more noise
Larger $\sigma$	Smooths more strongly and detects structures at a coarser scale

LoG is less sensitive to noise than an unsmoothed second derivative. It still requires a suitable scale and a suitable rule for accepting zero crossings.

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LoG Compared with Gradient Thresholding

Method	Edge evidence	Main strength	Main limit
[[Image Gradient and Edge Detection	Prewitt or Sobel]]	Large gradient magnitude	Simple and fast
LoG	Zero crossing of a smoothed second derivative	Smoothing and clear edge localization	Scale and zero-crossing acceptance must be selected
[[Canny Edge Detector	Canny]]	Thinned gradient maxima with hysteresis	Produces cleaner connected boundaries